The Sine Function

Section 13.4

The sine function, y = sin θ, matches the measure

θ of an angle in standard position with the y-

coordinate of a point on the unit circle.

This point is where the terminal side of the angle

intersects the unit circle.

sin0π 0π

sin 12

sinπ 03π

sin 12

sin 2π 0

The Sine Curve Graph

zero

maximum

zero zero

minimum

The graph on the last slide is called the parent

function of the sine function.

This means that:

1. it has not been stretched or compressed,

2. it has not been reflected over an axis,

3. it has not been moved up/down or right/left.

Today we will look at the first two

transformations.

The sine curve is a periodic function.

A periodic function repeats a pattern of y-values at

regular intervals.

One complete pattern is a cycle.

The period of a function is the horizontal length

of one cycle.

The period of the parent function sine curve is 2π.

Finding the Period of y = sin bθ

The period of a sine curve in the form y = sin bθ

is

b is the number of cycles that can be drawn from

0π and 2π.

2πp

b

Example

Find the period of the following sine curves and

tell how many cycles will be drawn from 0 to 2π:

a. y = sin 4θ

2π

4p

π

2p 4 cycles

b. y = sin 8θ

2π

8p

π

4p 8 cycles

θc. sin

3y

2π

1

3

p 6πp 1 cycle

3

Here is the graph for each of the problems from

the example.

a. y = sin 4θ

b. y = sin 8θ

θc. sin

3y

Amplitude of a Sine Curve

The amplitude of a periodic function is half the

difference between the maximum and minimum

values of the function.

The amplitude of the parent function sine curve

is 1.

Finding the Amplitude of y = a sin bθ

The amplitude of y = a sin bθ is a in the

equation.

Example

Find the amplitude and period for each sine.

Then write an equation for each curve

a.

2π

3p 2a

To find the equation for the sine curve, you must

find b. Remember b is the number of cycles from

0π to 2π.

The equation is y = a sin bθ.

3b

2sin3θy

b.

4πp 4a

4πp

2π4π

b

2πp

b

2π

4πb

1

2

14sin θ

2y

c.

2π

5p 3a 3sin5θy

Reflection Across the x-axis

If a in the equation y = a sin bθ is negative, then

the sine curve is reflected across the x-axis.

Example

Sketch one cycle of the graph of the sine curve.

a. y = 3 sin 2θ

1. The amplitude of the curve is 3. Mark this

on the y-axis.

2. The period is

2π

2p

πp

3. Divide the period into fourths to find the

hash marks for the graph.

4. The pattern for a sine curve is

zero-maximum-zero-minimum-zero

π

4

π

4

ππ

2

3π

4

b.

1. What is the amplitude of the sine curve?

amplitude = 2

2. What is the period?

period = 6π

θ2sin

3y

3. What would you divide x-axis into?

4. What is the pattern for a sine curve?

zero-maximum-zero-minimum-zero

6π

4

3π

2

3π

2

6π3π

9π

2

c. y = -4 sin 3θ

1. What is the amplitude of the sine curve?

amplitude = 4

2. What is the period?

2π

3p

3. What would you divide x-axis into?

4. What is the pattern for a sine curve?

Since there is a negative sign at the

beginning of the equation, we reflect the

graph over the x-axis.

zero-minimum-zero-maximum-zero

2π

3

4

π

6

π

6

2π

3

π

3

π

2